Functional Lifting 2.0: Efficient Convexifications for Imaging and Vision

One of the most versatile and powerful tools for solving computer vision and image processing problems is the class of energy-based methods. The basic idea is to associate with every conceivable solution a cost - or energy - in such a way that low energy values correspond to good solutions. Unfortunately, most practically relevant problems inherently lead to non-convex energy functions, so that the computation of global minimizers becomes very challenging. Consequently, the solution quality in classical local minimization methods heavily depends on the initialization and the particular choice of algorithm. In this project, we will develop global methods that rely on a convexification of non-convex energies in high-dimensional spaces by means of functional lifting in order to find globally optimal solutions. Existing approaches implemented this lifting by discretizing the range into labels, ultimately limiting the solution accuracy. In contrast, this project will extend the applicability of recent sublabel-accurate lifting methods that greatly reduce the extra cost of obtaining an approximate global solution and allow to gradually take into account the non-convexities while exploiting convex parts in the energy. This allows to develop solvers that scale gracefully with the non-convexity of the problem, and allow to efficiently compute close-to-globally optimal solutions on a wide range of optimization problems.